$\mathbb{Q}$-bonacci words and numbers

TL;DR

This paper introduces a generalization of multi-step Fibonacci numbers through a rational parameter, exploring binary words with specific factor constraints, and connects these to recurrence relations, compositions, Gray codes, and a new form of the golden ratio.

Contribution

It extends multi-step Fibonacci sequences to rational parameters, linking combinatorial word properties with recurrence relations and new mathematical constants.

Findings

Recurrence relations for rational q relate to restricted compositions.

Classical Fibonacci sequences are recovered when q is an integer.

Gray codes are discussed for the generalized words.

Abstract

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer we rediscover classical multi-step Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When $q$ is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio.